The theoretical backbone of this work is the
unbiased estimation of the Gaussian kernel by [1]}.
Based on Bochner's theorem [2]},
[1]} proposed random Fourier features to approximate a desired shift-invariant kernel. The method nonlinearly transforms a pair of vectors \({\mathbf {x}}\)  and \({\mathbf {y}}\)  using a random feature map \(\phi \) ;
the inner product between \(\phi ({\mathbf {x}})\)  and \(\phi ({\mathbf {y}})\) 
approximates the kernel evaluation on \({\mathbf {x}}\)  and \({\mathbf {y}}\) .
More precisely:
